A Weekly Digest of the Mathematical Internet

Tag Archives: numbers

Today we’re going to look at a few examples of going “up a dimension”. Our first example is what got me thinking about this theme. It’s a game called roTopo. (If you have trouble getting it to load, try using a different browser.)

Maybe you have played the game B-Cubed. RoTopo is similar—trace through a sequence of squares as they get eliminated one by one. I like B-Cubed because it combines spatial thinking with strategic thinking—planning ahead. Rotopo, with its twists and turns in 3D, stretches a player’s spatial thinking even further. I hope you enjoy giving it a try! Maybe you could design a roTopo level of your own with a drawing or with some blocks.

What else can we find when we look “up a dimension”? Maybe the most famous theorem in all of mathematics is the Pythagorean theorem. There are several ways we might try to take a^2+b^2=c^2 up a dimension. If we start to increase the numbers in the exponents, like a^3+b^3=c^3, we head in the direction of Fermat’s Last Theorem. If we add more terms, like a^2+b^2+c^2=d^2, we can find distances in 3D instead of 2D.

A right tetrahedron—the kind needed for de Gua’s Theorem.

And if those aren’t enough to make you go “wow”, then you need to hear about De Gua’s Theorem. The Pythagorean Theorem relates the sides of a right triangle. De Gua’s Theorem relates the faces of a right tetrahedron. The sum of the squares of the areas of the the three “leg” faces is equal to the square of the area of the “hypotenuse” face. So wild! You can read a proof de Gua’s Theorem here. The theorem is named for the 18th-century French mathematician who presented it to the Paris Academy of Sciences in 1783 (although it was known to others before him). De Gua’s Theorem in turn is a special case of a still more general theorem. Once mathematicians start upping dimensions, the sky is the limit!

Last up: Bibi-binary. No, that’s not the way that Justin Timberlake counts—although that funny thought is why I Googled “bibibinary” in the first place. But when I did, this totally silly number system popped up!

How to count in Bibi-binary.

Well, I guess it’s not the number system that’s silly so much, since it’s actually just hexadecimal. Hexadecimal is like binary, but up a couple of dimensions. The system uses sixteen symbols to represent numbers, just as the decimal system uses ten symbols and binary uses two. What makes Bibi-binary silly, then, is not its logical structure but how it sounds.

There are sixteen syllables in Bibi-binary, which are made from combinations of four consonants and four vowels. Three is “hi” and eight is “ko”. If you want to have three 16’s and eight more—56—that would be “hiko”. As another example, 66319344 is “hidihidihidiho”. Bibi-binary was invented in 1971 by a French singer and actor named Boby Lapointe.

I think it would be fun to learn to count in Bibi-binary. Can you believe that I could find zero (“ho”) videos online of people counting in Bibi-binary? I wonder if any of our readers might enjoy making one…

Hexadecimal is not just fun and games. It’s also used for making codes to stand for colors, especially in making webpages. Most of Math Munch is either 683D29 or 6AB690, would you believe. You can explore using hexadecimal to name colors in this applet.

You can learn lots more about Bibi-binary on the great website dCode, and you’ll also find an applet there that can convert between decimal and Bibi-binary. DCode has lots of tools related to cryptography (get it?) and other math topics, too.

Do you have any favorite examples of math that goes “up a dimension”? We’d love to hear about them in the comments.

There’s a huge celebration here in Princeton—in part because Pi Day is also Albert Einstein’s birthday, and Albert lived in Princeton for the last 22 years of his life. One event involves kids reciting digits of pi and and is hosted by John Conway and his son, a two-time winner of the contest. I’m looking forward to attending! But as has been noted, memorizing digits of pi isn’t the most mathematical of activities. As Evelyn Lamb relays,

I do feel compelled to point out that besides base 10 being an arbitrary way of representing pi, one of the reasons I’m not fond of digit reciting contests is that, to steal an analogy I read somewhere, memorizing digits of pi is to math as memorizing the order of letters in Robert Frost’s poems is to literature. It’s not an intellectually meaningful activity.

I haven’t memorized very many digits of pi, but I have memorized a digit of pi that no one else has. Ever. In the history of the world. Probably no one has ever even thought about this digit of pi.

And you can have your own secret digit, too—all thanks to Simon Plouffe‘s amazing formula.

Simon’s formula shows that pi can be calculated chunk by chunk in base 16 (or hexadecimal). A single digit of pi can be plucked out of the number without calculating the ones that come before it.

The discovery of this formula came as a surprise. For centuries it had been assumed that there was no way to compute the nth digit of π without calculating all of the preceding n − 1 digits.

Check out some of Simon’s math art!

Simon is a mathematician who was born in Quebec. In addition to his work on the digits of irrational numbers, he also helped Neil Sloane with his Encyclopedia of Integer Sequences, which soon online and became the OEIS (previously). Simon is currently a Trustee of the OEIS Foundation.

There is a wonderful article by Simon and his colleagues David Bailey, Jonathan Borwein, and Peter Borwein called The Quest for Pi. They describe the history of the computation of digits of pi, as well as a description of the discovery of their digit-plucking formula.

According to the Guinness Book of World Records, the most digits that someone has memorized and recited is 67,890. Unofficial records go up to 100,000 digit. So just to be safe, I’ve used an algorithm by Fabrice Bellard based on Simon’s formula to calculate the 314159th digit of pi. (Details here and here.) No one in the world has this digit of pi memorized except for me.

Ready to hear my secret digit of pi? Lean in and I’ll whisper it to you.

The 314159th digit of pi is…7. But let’s keep that just between you and me!

And just to be sure, I used this website to verify the 314159th digit. You can use the site to try to find any digit sequence in the first 200 million digits of pi.

Aziz & Peter’s patterns.

Next up: we met Aziz Inan in last week’s post. This week, in honor of Pi Day, check out some of the numerical coincidences Aziz has discovered in the early digits in pi. Aziz and his colleague Peter Osterberg wrote an article about their findings. By themselves, these observations are nifty little patterns. Maybe you’ll find some more of your own. (This kind of thing reminds me of the Strong Law of Small Numbers.) As Aziz and Peter note at the end of the article, perhaps the study of such little patterns will one day help to show that pi is a normal number.

And last up this week, to get your jam on as Saturday approaches, here’s the brand new Pi Day Anthem by the recently featured John Sims and the inimitable Vi Hart.

Welcome to this week’s Math Munch! And, if you’re a student or teacher, welcome to a new school year!

One of the most exciting events in the world of math happened this August– the awarding of the Fields Medal! This award honors young mathematicians who have already done awesome mathematical work and who show great promise for the future. It also only happens every four years, at the beginning of an important math conference called the International Congress of Mathematicians, so it’s a very special occasion when it does!

Maryam Mirzakhani, first woman ever to win a Fields Medal

This year’s award was even more special than usual, though. Not only were there four winners (more than the usual two or three), but one of the winners was a woman!

Now, if you’re like me, you probably heard about the Fields Medal and thought, “There’s no way I’ll understand the math that these Field Medalists do.” But this couldn’t be more wrong! Thanks to these great articles from Quanta Magazine, you can learn a lot about the super-interesting math that the Fields Medalists study– and why they study it.

Manjul Bhargava

One thing you’ll immediately notice is that each Fields Medalist has non-math interests that inspire their mathematical work. Take Manjul, for instance. When he was a kid, his grandfather introduced him to Sanskrit poetry. He was fascinated by the patterns in the rhythms of the poems, and the number patterns that he found inspired him to study the mathematics of number patterns– number theory!

But, don’t just take my word for it– you can read all about Manjul and the others in these great articles! And did I mention that they come with videos about each mathematician?

Want to read more about this year’s Fields Medallists? Check out Alex Bellos’s article in The Guardian. Which brings me to…

… What’s your favorite number? Is it 7? If it is, then you’re in good company! Alex polled more than 30,000 people about their favorite number, and the most popular was 7. But why? What’s so special about 7? Here’s why Alex thinks 7 is such a favorite:

Why do you like your favorite number? People gave Alex all kinds of different reasons. One woman said about 3, her favorite number, “3 wishes. On the count of 3. 3 little pigs… great triumvirates!” Alex made these questions the topic of the first chapter of his new book, The Grapes of Math. (Get the reference?) In this book, Alex shares many curious ways that math appears in our world. Did you know that a weird pattern in numbers can be used to catch criminals? Or that the Game of Life, a simple computer program, shares surprisingly many characteristics with real life? These are only a few of the hundreds of topics Alex covers in his book. Whether you’re a math whiz or a newbie, you’ll learn something new on every page.

Alex currently writes about math for The Guardian in a blog called, “Alex’s Adventures in Numberland”— but he also loves and writes about soccer (or futbol, as it’s called in his native Brazil)! He even wrote a few articles for his blog about math and soccer.

Do you have any questions for Alex? (About math, soccer, or their intersection?) Write them here and you might find them featured in our interview with Alex!

Good writing about math is hard to find. If you’ve ever picked up a standard math textbook, you’ll know what I mean. But reading something fascinating, that grabs your interest from the first page and leads you through the most complex ideas like they’re as natural as anything you’ve observed, is a great way to learn. The Grapes of Math and “Alex’s Adventures in Numberland” do just that. Give them a go!

April is Mathematics Awareness Month. So happy Mathematics Awareness Month! This year’s theme is “Mathematics, Magic, and Mystery”. It’s inspired by the fact that 2014 would have marked Martin Gardner’s 100th birthday.

A few of the mathy morsels that await you this month on mathaware.org!

Each day this month a new piece of magical or mysterious math will be revealed on the MAM site. The mathematical offering for today is a card trick that’s based on the Fibonacci numbers. Dipping into this site from time to time would be a great way for you to have a mathy month.

It is white’s turn to move. Who will win this Hexapawn game?

Speaking of Martin Gardner, I recently ran across a version of Hexapawn made in the programming language Scratch. Hexapawn is a chess mini-game involving—you guessed it—six pawns. Martin invented it and shared it in his Mathematical Games column in 1962. (Here’s the original column.) The object of the game is to get one of your pawns to the other side of the board or to “lock” the position so that your opponent cannot move. The pawns can move by stepping forward one square or capturing diagonally forward. Simple rules, but winning is trickier than you might think!

The program I found was created by a new Scratcher who goes by the handle “puttering”. On the site he explains:

I’m a dad. I was looking for a good way for my daughters to learn programming and I found Scratch. It turns out to be so much fun that I’ve made some projects myself, when I can get the computer…

puttering’s Scratch version of Conway’s Game of Life

Something that’s super cool about puttering’s Hexapawn game is that the program learns from its stratetgy errors and gradually becomes a stronger player as you play more! It’s well worth playing a bunch of games just to see this happen. puttering has other Scratch creations on his page, too—like a solver for the Eight Queens puzzle and a Secret Code Machine. Be sure to check those out, too!

Last up, our friend Nalini Joshi recently travelled to a meeting of the Australian Academy of Science, which led to a little number puzzle.

What unusual ways of describing a number! Trying to learn about these terms led me to an equally unusual calculator, hosted on the Math Celebrity website. The calculator will show you calculations about the factors of a numbers, as well as lots of categories that your number fits into. Derek Orr of Math Year-Round and I figured out that Nalini’s clues fit with multiple numbers, including 185, 191, and 205. So we needed more clues!

Can you find another number that fits Nalini’s clues? What do you think would be some good additional questions we could ask Nalini? Leave your thoughts in the comments!

A result from the Number Property Calculator

I hope this post helps you to kick off a great Mathematics Awareness Month. Bon appetit!

What does pi look like? The first 10,000 digits of pi, each digit 0 through 9 assigned a different color.

You probably know some pretty cool things about the number pi. Perhaps you know that pi has quite a lot to do with circles. Maybe you know that the decimal expansion for pi goes on and on, forever and ever, without repeating. Maybe you know that it’s very likely that any string of numbers– your birthday, phone number, all the birthdays of everyone you know listed in a row, followed by all their phone numbers, ANYTHING– can be found in the decimal expansion of pi.

But did you know that pi can be approximated by dropping needles on a piece of paper? Well, it can! If you drop a needle again and again on a lined piece of paper, and the needle is the same length as the distance between the lines, the probably that the needle lands on a line is two divided by pi. This experiment is called Buffon’s needle, after the French naturalist Buffon.

If the angle the needle makes with the lines is in the gray area (like the red needle’s angle is), it will cross the line. If the angle isn’t, it won’t. The possible angles trace out a circle. The closer the center of the needle (or center of the circle) is to the line, the larger the gray area– and the higher the probability of the needle hitting the line.

This may seem strange to you– but if you think about how the needle hitting a line has a lot to do with the distance between the middle of the needle and the nearest line and the angle it makes with the lines, maybe you’ll start to think about circles… and then you’ll get a clue about the connection between this experiment and pi. Working out this probability exactly requires some pretty advanced mathematics. (Feeling ambitious? Read about the calculation here.) But, you can get some great experimental results using this Buffon’s needle applet.

Click on the picture to try the applet.

I had the applet drop 500 needles. Then, the applet used the fact that the probability of the needle hitting a line should be two divided by pi and the probability it measured to calculate an approximation for pi. It got… well, you can see in the picture. Pretty close, right?

Here’s another thing you might not know: pi is a transcendental number. Sounds trippy– but, like some other famous numbers with letter names, like e, pi can never be the solution to an algebraic equation involving whole numbers. That means that no matter what equation you give me– no matter how large the exponent, how many negatives you toss in, how many times you multiply or divide by a whole number– pi will never, ever be a solution. Maybe this doesn’t sound amazing to you. If not, check out this video from Numberphile about transcendental numbers. Numbers like pi and e don’t do mathematical things we’re used to numbers doing… and it’s pretty weird.

Still curious about transcendental numbers? Here’s a page listing the fifteen most famous transcendental numbers. My favorite? Definitely the fifth, Liouville’s number, which has a 1 in each consecutive factorial numbered place.

Finally, maybe you don’t like pi. Maybe you like cookies instead. Lucky for you, you can do many mathematical things with cookies, too. Like make cookie tessellations! This mathematical artist and baker made cookie cutters in the shapes of tiles from Escher tessellations and used them to make mathematical cookie puzzles. Beautiful, and certainly delicious.

If you happen to have a 3D printer, you can make your own Escher cookie cutters. Here’s a link to print out the lizard cutter. If you don’t have a 3D printer, you could try printing out a 2D image of an Escher tessellation and tracing a tile onto a sheet of paper. Cut out the tile, roll out your dough, and slice around the outside of the tile to make your cookies. If you do it right, you shouldn’t have to waste any dough…

On his website, Mariano calls himself the “Wonder Injector,” a “writer of science with the mission of the magician.” And that video certainly looked like magic! I wonder how the trick works…

Mariano’s website is full of fun videos involving mathe-magical tricks. I like watching them, being completely baffled, and then figuring out how the trick works. Here’s another one that I really like, about a fictional plane saved from crashing. It’s a little creepy.

How does this trick work???

Next up is one of my favorite number pattern — Pascal’s Triangle. Pascal’s Triangle appears all over mathematics– from algebra to combinatorics to number theory.

Pascal’s Triangle always starts with a 1 at the top. To make more rows, you add together two numbers next to each other and put their sum between them in the row below. For example, see the two threes beside each other in the fourth row? They add to 6, which is placed between them in the fifth row.

Pascal’s Triangle is full of interesting patterns (what can you find?)– but my favorite patterns appear when you color the numbers according to their factors.

That’s just what Brent Yorgey, computer programmer and author of the blog “The Math Less Travelled,” did! Here’s what you get if you color all of the numbers that are multiples of 2 gray and all of the numbers that aren’t multiples of 2 blue.

Recognize that pattern? It’s a Sierpinski triangle fractal!

If you thought that was cool, check out this one based on what happens if you divide all the numbers in the triangle by 5. The multiples of 5 are gray; the numbers that leave a remainder of 1 when divided by 5 are blue, remainder 2 are red, remainder 3 are yellow, and remainder 4 are green. And here’s one based on what happens if you divide all the numbers in the triangle by 6.

See the yellow Sierpinski triangle below the blue, red, green, and purple pattern? Why might the pattern for multiples of two appear in the triangle colored based on multiples of 6?

Step 3: Write about one of the previous situations using as many
of these words as possible. Try to avoid referencing the situation
directly. Write no more than seven words per line.

Here’s one that I like:

ASPARAGUS, by Sarah Goldfarb

An infinity of hunger within me
Dividing a bunch of green
Snap and sizzle,
Green parentheses in a pan
The aromatic property
Simplifying my want
Producing a need
Each fraction of a second
Dragging its feet impatiently as I wait
And when it is distributed on my plate
It is only a moment before zero
Units of nourishment remain.

Maybe you’ll try writing a poem of your own! If you do, we’d love to see it.

The letters in the table are the abbreviations of the chemical elements—like gold, helium, and iron—that are found on the usual periodic table. With a little creativity, they can also be abbreviations for the names of a bunch of celebrated mathematicians. Clicking on a square brings up the mathematician’s biography. I like guessing who might pop up!

In 2000, Erik worked with Robert Connelly and Günter Rote to show that any wound-up 2D shape made of hinged sticks can be unwound without breaking, crossing, or lifting out of the plane. In the end, the shape must be convex, so that it doesn’t have any dents in it. For a while Erik and his colleagues thought that some linkages might be “locked” and unwinding some of the examples they created took months. You can find some great animations shared on the webpage that describes their result that locked linkages don’t in fact exist.

One thing that amazes me about Erik’s mathematical work is how young the problems are that he works on and solves. You might think a problem that can be put in terms of such simple ideas would have been around for a while, but in fact this problem of unwinding linkages was first posed only in the 1970s! It just goes to show that there are new simple math problems just waiting to be invented all the time.

Finally, I was so glad to run across this short film called Dance Squared. It was made by René Jodoin, a Canadian director and producer. Check out how much René expresses with just a simple square!